Centola & Macy (2007) Complex contagions and the weakness of long ties

Centola D, Macy M (2007) Complex contagions and the weakness of long ties. American Journal of Sociology 113:702-734. 

Some contagions like job information or disease spread through simple contact.

However, “many collective behaviors also spread through social contact, but when these behaviors are costly, risky, or controversial, the willingness to participate may require independent affirmation or reinforcement from multiple sources. We call these ‘complex contagions’ because successful transmission depends upon interaction with multiple carriers.” (p 703)

Granovetter (1973) used the phrase “weak ties” to indicate two possible things. The first is relational: weak ties are between individuals who see each other infrequently, who are not closely bonded, who don’t have very strong levels of trust (as opposed to strong ties, in which the opposite is true). The second is structural: at the population level, weak ties are those that connect otherwise highly clustered neighborhoods. Centola and Macy focus on these, and call them “long ties.”

The strength of long ties in simple contagion

• Long ties facilitate diffusion by providing shortcuts (Granovetter)
• Only a very small fraction of long ties can give even highly clustered networks high “degrees of separation,” which means that information and disease can spread rapidly even in a “small world” (Watts & Strogatz)

Early “threshold” models of contagion (Granovetter 1978, Schelling 1978) looked at the number of initial seeds required to tip a population-wide cascade.

Complex contagions

• “While all contagions have a minimum threshold of one, the range of nonzero thresholds can be quite large.” (p 706)
• “The distinction between simple and complex refers to the number of sources of exposure required for activation, not the number of exposures. A contagion is complex if its transmission requires an individual to have contact with two or more sources of activation.” (p 707)

Mechanisms of complex contagion

There are at least four social mechanisms that might explain why complex contagions require multiple sources of activation.

1. Strategic complementarity. Adopting an innovation may be costly, especially for early adopters. The costs and benefits for investing in a public good often depends on the number of prior contributors (the “critical mass”).
2. Credibility. Innovations often lack credibility until adopted by neighbors.
3. Legitimacy. Innovators risk being shunned as deviants until there is a critical mass of early adopters.
4. Emotional contagion. There are expressive and symbolic impulses in human behavior that can be communicated and amplified in spatially and socially concentrated gatherings (ala echo chambers).

This paper tests the previously held assumption that network properties conducive to the spread of disease and information (simple contagions) are also conducive to the spread of complex contagions.

Summary of findings:

• We show that for complex contagions, long ties can be weak in both of Granovetter’s meanings, structural as well as relational.
• The implication of the relational meaning is immediately apparent. A low level of trust and familiarity between socially distant persons means the relationship is weak, and this inhibits the ability of one person to influence the other.
• What is not at all obvious is that long ties can also have a structural weakness—they are nontransitive. For the spread of information, transitive ties between friends tend to be redundant, such that we hear the same thing from multiple friends. However, when activation requires confirmation or reinforcement from two or more sources, the transitive structure that was redundant for the spread of information now becomes an essential pathway for diffusion.
• Thus, while weak ties are beneficial for the spread of new information precisely because they are nonredundant, for complex contagions uniqueness becomes a weakness rather than a strength.

Modeling

Bridges: A “local bridge” links otherwise disjoint neighborhoods (i.e., no members in common). A local bridge makes a connection shorter. The analysis considers these local bridges, and refers to them as “bridges.”

If a connection requires multiple contacts, then a bridge must consist of multiple ties. Thus a bridge can be measured both by its length (the range that is spanned) and by its width (the number of ties it contains).

A Ring Lattice

They start with the small world network algorithm from Watts and Strogatz 1998. These results are all analytical.

Good point: There is a difference between thresholds expressed as either the number or the fraction of activated neighbors. Fractional thresholds hold implicit assumptions about the influence of nonadopters – i.e., nonadopters hold influence, but in the opposite direction. If nonadopters exert countervailing influence, then as neighborhood size increases, a greater number of neighbors are required to trigger adoption.

They represent the threshold as a fraction with both numerator and denominator explicitly stated – the number of activated nodes as a fraction of the total number of neighbors.

Summary of ring lattice results:

• While a single random tie is sufficient to promote the spread of simple contagions, complex contagions require more rewiring in order to benefit from randomization. The number of ties that need to be randomly rewired increases exponentially with the number required to form a bridge, which in turn increases exponentially with the required number of activated neighbors.
• As the ring becomes increasingly randomized, the width of the bridges that make up the lattice structure may be eroded below the critical width required for the contagion to spread. For complex contagions, there is a critical upper limit of randomization, above which it cannot propagate.

Long ties on higher dimensional networks.

Computer simulations on two-dimensional lattices with Moore neighborhoods.

Keeping network density constant (by adjusting the size), they experimented with two neighborhood sizes, z = 8 (Moore r = 1) and z = 48 (Moore r = 3). Looking a activation thresholds up to the upper limit for complex contagion (a = 3 for z = 8, a = 19 for z = 48). A proportion p of the links were randomly rewired (so p = 0 is a regular lattice and p = 1 is a random network).

For simple contagion, only a small fraction of random ties are needed to allow propagation rate to approach those on a random graph. However, this small world effect does not generalize to complex contagions.

“The results for Moore neighborhoods show that random ties do not help complex contagions at very low and very high levels of randomization.”

Three principle results:

1. Complex contagions fail to benefit from low levels of randomization.
2. Increasing p has a nonmonotonic effect on complex contagions, exhibiting a U-shaped effect, in which randomization starts to help—but ultimately impedes—propagation.
3. As p exceeds a critical upper limit, complex contagions entirely fail to propagate.
• As p increases, there is an abrupt phase transition from near-complete success on each trial to near-complete failure for the contagion to cascade.

“Watts and Strogatz (1998) discovered that simple contagions could spread as fast on a highly clustered small world network as on a more randomized topology. This was important because social networks tend to be highly clustered and rarely (if ever) random. Figures 3 and 4 reveal that this “small world effect” becomes more pronounced as thresholds increase slightly above the level of simple contagions. That is, complex contagions with relatively low thresholds can actually spread faster on a highly clustered small world network than on either a network that is more random or one that is more clustered. However, as thresholds get higher still, the small world effect disappears entirely. In short, the essential difference between simple and complex contagions can be distilled as follows. For simple contagions, too much clustering means too few long ties, which slows down cascades. For complex contagions, too little clustering means too few wide bridges, which not only slows down cascades but can prevent them entirely.” (p 723)

Robustness checks:

• Threshold heterogeneity
• Used a Gaussian distribution of thresholds.
• Also tried randomly changing within-node thresholds.
• Also used a stochastic activation based on a logistic function related to the proportion of neighbors.
• Qualitatively same results under all conditions.
• Heterogeneity of influence
• Created status differences: high and low-status neighbors. High-status nodes could activate their neighbors without the need for additional sources. Randomly assigned 1/8 of all nodes to be “high-status.”
• Introducing a small fraction of high status nodes does not mitigate the need for wide bridges. There was no improvement in the propagation of complex contagions as p increased. This was in part due to an artifact – the influence of low-status nodes was reduced to keep the mean influence constant, so that it now took more low status nodes for activation.
• However, as the number of high-status nodes increases, the propagation of complex contagions can begin to resemble that for simple contagions.
• Strong and weak ties
• Used a simple contagion model, but assigned nodes involved in a closed triad to be “close friends” with weight 1.0, and those in an open triad to be “acquaintance” with weight 0.5 (thus requiring two of them, making it equivalent to complex contagion). The results were similar to the complex contagion stuff.
• Scale free (Barabasi-Albert) network
• The low degree in most nodes precluded most contagions from spreading with a > 2, even with p = 0.
• For thresholds of 2/z, the results were similar to the regular lattice.
• A scale-free network can be even more sensitive to perturbation than a regular lattice.

Discussion

• Network topologies that make it easy for everyone to know about something do not necessarily make it likely that people will change their behavior.
• Having friends in a community who are also friends with one another increases the probability of joining, compared to friends in a community who do not know one another. This suggests that the growth of a community depends less on weak ties that span longer social distances and more on wide bridges.
• A consistent finding in social movement research is that participation spreads most effectively in populations that are spatially clustered, such as ethnic enclaves.
• The optimal topology for the spread of collective action may depend on the costs and risks of participation and thus on the relative importance of information versus social reinforcement in mobilizing action. Resolving the coordination dilemma then requires multiple contacts who reinforce both the credibility of the information and the normative importance of taking action.
• Where public health innovations contravene existing social norms, health reform is likely to require social reinforcement, not simply access to information.